L(s) = 1 | + 6·2-s + 11·4-s − 30·7-s − 36·8-s + 53·9-s + 78·13-s − 180·14-s − 267·16-s + 318·18-s + 468·26-s − 330·28-s + 288·29-s − 558·32-s + 583·36-s − 606·37-s − 222·47-s − 11·49-s + 858·52-s + 1.08e3·56-s + 1.72e3·58-s + 752·61-s − 1.59e3·63-s + 895·64-s − 72·67-s − 1.90e3·72-s − 2.19e3·73-s − 3.63e3·74-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 11/8·4-s − 1.61·7-s − 1.59·8-s + 1.96·9-s + 1.66·13-s − 3.43·14-s − 4.17·16-s + 4.16·18-s + 3.53·26-s − 2.22·28-s + 1.84·29-s − 3.08·32-s + 2.69·36-s − 2.69·37-s − 0.688·47-s − 0.0320·49-s + 2.28·52-s + 2.57·56-s + 3.91·58-s + 1.57·61-s − 3.17·63-s + 1.74·64-s − 0.131·67-s − 3.12·72-s − 3.52·73-s − 5.71·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.808167926\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.808167926\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 p T + p^{3} T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - 3 T + p^{3} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 53 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 358 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7801 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 13682 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 1910 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 144 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10114 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 303 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 100978 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 149605 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 111 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 138274 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 376 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 588373 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 1098 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 830 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 438 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1218094 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 852 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.87106442819228722157706335898, −11.07995434205870930926470334777, −10.41203109147594006694007539688, −10.11178553201213220270499347407, −9.620028170308798122243178855427, −9.129556173315592289881134145777, −8.662179121947815531624380167550, −8.183092059244316681605186760341, −7.11962844120745145542551012033, −6.62160625949662040169102810198, −6.52403135380960089224769454620, −5.98496614713293216223272105786, −5.28320919903726654455402996808, −4.80221803568983395112351523134, −4.22622582134335857619391331233, −3.75254845520954672112358233845, −3.38951864799546477142079900616, −2.91069101221838435288395217451, −1.68413095921178438608396351490, −0.57254729840677549771603016597,
0.57254729840677549771603016597, 1.68413095921178438608396351490, 2.91069101221838435288395217451, 3.38951864799546477142079900616, 3.75254845520954672112358233845, 4.22622582134335857619391331233, 4.80221803568983395112351523134, 5.28320919903726654455402996808, 5.98496614713293216223272105786, 6.52403135380960089224769454620, 6.62160625949662040169102810198, 7.11962844120745145542551012033, 8.183092059244316681605186760341, 8.662179121947815531624380167550, 9.129556173315592289881134145777, 9.620028170308798122243178855427, 10.11178553201213220270499347407, 10.41203109147594006694007539688, 11.07995434205870930926470334777, 11.87106442819228722157706335898